It may seem disturbing that such a simple, passive, physically
rigorous simulation of a mass-springoscillator should have to make
use of state variables which grow without bound for the limiting cases
of simple harmonic motion at frequencies zero and half the sampling
rate. This is obviously a valid concern in practice as well.
However, it is easy to show that this only happens at dc and
,
and that there is a true degeneracy at these frequencies, even in the
physics. For all frequencies in the audio range (e.g., for typical
sampling rates), such state variable growth cannot occur. Let's take
closer look at this phenomenon, first from a signal processing point
of view, and second from a physical point of view.